Eccentric poet to take on, becoming a handicap (14)
I believe the answer is:
cambridgeshire
'a handicap' is the definition.
(Cambridgeshire Handicap is a horse race)
'eccentric poet to take on' is the wordplay.
'eccentric' becomes 'cam' (eccentric cam is a type of mechanical part).
'poet' becomes 'bridges' (Robert Bridges, 1844-1930).
'to take on' becomes 'hire' (both can mean to employ).
'cam'+'bridges'+'hire'='CAMBRIDGESHIRE'
'becoming' acts as a link.
(Other definitions for cambridgeshire that I've seen before include "autumn race" , "over a thousand square miles" .)